Numerical solution for the fluid flow between active elastic walls

Abstract

We analyse a model of the fluid flow between elastic walls simulating arteries actively interacting with the blood. The lubrication theory for the flow is coupled with the pressure and shear stress from the walls. The resulting nonlinear partial differential equation describes the displacement of the walls as a function of the distance along the flow and time. The equation is solved numerically using the one-dimensional integrated radial basis function network method. A solution in the form of a self-sustained train of pulses is obtained. Numerical experiments demonstrate the process of formation of the train from randomly chosen initial conditions. Dependence of the pulses on the boundary conditions is explored. References

• S. J. Sherwin, L. Formaggia, J. Peiro and V. Franke. Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int. J. Numer. Meth. Fluids, 43:673–700, 2003. doi:10.1002/fld.543
• C. Kleinstreuer. Biofluid Dynamics. Taylor and Francis, 2006.
• M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng., 28:1281–1299, 2000. doi:10.1114/1.1326031
• K. S. Matthys, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, P. R. Verdonck, K. H. Parker and S. J. Sherwin. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech., 40:3476–3486, 2007. doi:10.1016/j.jbiomech.2007.05.027
• A. J. Roberts. A One-dimensional Introduction to Continuum Mechanics. World Scientific. Singapore, 1994. doi:10.1142/2496
• D. V. Strunin, Fluid flow between active elastic plates. ANZIAM J., 50:C871–C883, 2009. doi:10.21914/anziamj.v50i0.1452
• D. V. Strunin, Autosoliton model of the spinning fronts of reaction. IMA J. Appl. Math., 63:163–177, 1999. doi:10.1093/imamat/63.2.163
• R. Huang and Z. Suo. Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys., 91:1135–1142, 2002. doi:10.1063/1.1427407
• J. King. The isolation oxidation of silicon: The reaction-controlled case. SIAM J. Appl. Math. 49:1064–1080, 1989. doi:10.1137/0149064
• D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. Local moving least square-one-dimensional integrated radial basis function networks technique for incompressible viscous flows. Int. J. Numer. Meth. Fluids, 70:1443–1474, 2012. doi:10.1002/fld.3640
• P. Le, N. Mai-Duy, T. Tran-Cong and G. Baker. A meshless modeling of dynamic strain localization in quasi-brittle materials using radial basis function networks, CMES–Comp. Model. Eng., 25:43–66, 2008. doi:10.3970/cmes.2008.025.043
• N. Mai-Duy and T. Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks, Neural Networks, 14:185–199, 2001. doi:10.1016/S0893-6080(00)00095-2